On the existence of stable rotation numbers for monotone recurrence relations

This paper aims to establish various criteria for the existence of stable rotation numbers, motivated by their relationship with positive topological entropy of the system generated by the monotone recurrence relation (Equation1(1) Δ(xn−k,…,xn,…,xn+l)=0 for all n∈Z,(1) ). By means of depinning forces and Birkhoff ordered circles for particle systems, we obtain that if (Equation1(1) Δ(xn−k,…,xn,…,xn+l)=0 for all n∈Z,(1) ) has a subsolution and a supersolution exchanging rotation numbers ω1 and ω2, then (ω1,ω1) consists of stable rotation numbers. Relying on it, we further give a necessary and sufficient condition for the existence of irrational stable rotation numbers, showing the equivalence to the presence of Angenent's rotation number exchange. We then define the Birkhoff connected set as a compact invariant set where any two elements admit a Birkhoff connection. It is demonstrated that if the rotation set of a Birkhoff connected set is not a singleton, then any element in the interior of the convex hull of the rotation set is a stable rotation number for (Equation1(1) Δ(xn−k,…,xn,…,xn+l)=0 for all n∈Z,(1) ).